## On the maximum value of the first coefficients of Kazhdan-Lusztig polynomials for symmetric groups

J. Math. Sci. Univ. Tokyo
Vol. 1 (1994), No. 2, Page 461--469.

Tagawa, Hiroyuki
On the maximum value of the first coefficients of Kazhdan-Lusztig polynomials for symmetric groups
In this article, we show that max$\{c^-(w);w \in \frak S_n\} = [n^2/4]$, where $c^-(w)$ is the number of elements covered by $w \in \frak S_n$ in the Bruhat order. Using this result, we can see that the maximum value of the first coefficients of Kazhdan-Lusztig polynomials for $\frak S_n$ equals $[n^2/4]- n + 1$.